An importation inference. (Moved into main set.mm as imp5g583
and may be
deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff
Hankins, 7-Jul-2009.) (Proof modification is discouraged.)
(New usage is discouraged.)

An importation inference. (Moved into main set.mm as imp55584
and may be
deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by Jeff
Hankins, 7-Jul-2009.) (Proof modification is discouraged.)
(New usage is discouraged.)

An importation inference. (Moved into main set.mm as imp511585 and may
be deleted by mathbox owner, JGH. --NM 29-May-2014.) (Contributed by
Jeff Hankins, 7-Jul-2009.) (Proof modification is discouraged.)
(New usage is discouraged.)

A condition which implies existential uniqueness. (Moved into main
set.mm as eqeu2936 and may be deleted by mathbox owner, JGH.
--NM
29-May-2014.) (Contributed by Jeff Hankins, 8-Sep-2009.)
(Proof modification is discouraged.) (New usage is discouraged.)

An image under the converse of a restriction. (Contributed by Jeff
Hankins, 12-Jul-2009.) (Moved to cnvresima5159 in main set.mm and may be
deleted by mathbox owner, JGH. --NM 23-Dec-2013.)
(Proof modification is discouraged.) (New usage is discouraged.)

A closed interval with identical lower and upper bounds is a singleton.
(Moved into main set.mm as iccid10696 and may be deleted by mathbox owner,
JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 13-Jul-2009.)
(Proof modification is discouraged.) (New usage is discouraged.)

If the upper bound of one open interval is less than or equal to the
lower bound of the other, the intervals are disjoint. (Moved into main
set.mm as ioodisj10760 and may be deleted by mathbox owner, JGH.
--NM
29-May-2014.) (Contributed by Jeff Hankins, 13-Jul-2009.)
(Proof modification is discouraged.) (New usage is discouraged.)

Cancel equal divisors in a division. (Contributed by Jeff Hankins,
29-Sep-2013.) (Moved to divcan79464 in main set.mm and may be deleted by
mathbox owner, JGH. --NM 21-May-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)

If a nonempty set of real numbers has a lower bound, its infimum is less
than or equal to any of its elements. (Contributed by Jeff Hankins,
15-Sep-2013.) (Moved to infmrlb9730 in main set.mm and may be deleted by
mathbox owner, JGH. --NM 21-May-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)

A set is open in the standard topology of the reals precisely when every
point can be enclosed in an open ball. (Contributed by Jeff Hankins,
23-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)

A set is open in the standard topology of the reals precisely when every
point can be enclosed in an arbitrarily small ball. (Contributed by
Jeff Hankins, 22-Sep-2013.) (Proof shortened by Mario Carneiro,
30-Jan-2014.)

One nonzero integer divides another integer if and only if their
quotient is an integer. (Moved to cnvresima5159 in main set.mm and may be
deleted by mathbox owner, JGH. --NM 28-Feb-2014.) (Contributed by Jeff
Hankins, 29-Sep-2013.) (Proof modification is discouraged.)
(New usage is discouraged.)

Two equal reduced fractions have the same numerator and denominator.
(Contributed by Jeff Hankins, 29-Sep-2013.) (Moved into main set.mm as
qredeq12780 and may be deleted by mathbox owner, JGH.
--NM 13-Oct-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)

Every rational number has a unique reduced form. (Contributed by Jeff
Hankins, 29-Sep-2013.) (Moved into main set.mm as qredeq12780 and may be
deleted by mathbox owner, JGH. --NM 13-Oct-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)

An alternate definition of connectedness. (Moved into main set.mm as
dfcon217140 and may be deleted by mathbox owner, JGH.
--NM 29-May-2014.)
(Contributed by Jeff Hankins, 9-Jul-2009.) (Revised by Mario Carneiro,
8-Apr-2015.) (Proof modification is discouraged.)
(New usage is discouraged.)

Two equivalent ways of saying that a subspace topology is connected.
(Moved into main set.mm as connsub17142 and may be deleted by mathbox
owner, JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins,
9-Jul-2009.) (Proof modification is discouraged.)
(New usage is discouraged.)

A subset of the reals is connected iff it has the interval property.
(Moved into main set.mm as reconn18328 and may be deleted by mathbox owner,
JGH. --NM 29-May-2014.) (Contributed by Jeff Hankins, 15-Jul-2009.)
(Proof modification is discouraged.) (New usage is discouraged.)

Corollary of reconn18328. The set of real numbers is connected.
(Moved
into main set.mm as retopcon18329 and may be deleted by mathbox owner,
JGH.
--NM 29-May-2014.) (Contributed by Jeff Hankins, 17-Aug-2009.)
(Proof modification is discouraged.) (New usage is discouraged.)

A closed interval is connected. (Moved into main set.mm as iccconn18330
and may be deleted by mathbox owner, JGH. --NM 29-May-2014.)
(Contributed by Jeff Hankins, 17-Aug-2009.)
(Proof modification is discouraged.) (New usage is discouraged.)

The predicate "
is finer than ."
This property is, in a
sense, the opposite of refinement, as refinement requires every element
to be a subset of an element of the original and fineness requires that
every element of the original have a subset in the finer cover
containing every point. I do not know of a literature reference for
this. (Contributed by Jeff Hankins, 28-Sep-2009.)

The property of being a refinement of a cover. Dr. Nyikos once
commented in class that the term "refinement" is actually
misleading and
that people are inclined to confuse it with the notion defined in
isfne25679. On the other hand, the two concepts do
seem to have a dual
relationship. (Contributed by Jeff Hankins, 18-Jan-2010.)